Abstract

A more fundamental concept than the minimal residual method is proposed in this paper to solve an $n$-dimensional linear equations system ${\bf A}{\bf x}={\bf b}$ in an $m$-dimensional Krylov subspace. We maximize the orthogonal projection of ${\bf b}$ onto ${\bf y}$: $={\bf A}{\bf x}$. Then, we can prove that the maximal projection solution (MP) is better than that obtained by the least squares solution (LS) with $\|{\bf b}-{\bf A}{\bf x}_{\mbox{\scriptsize MP}}\|<\|{\bf b}-{\bf A}{\bf x}_{\mbox{\scriptsize LS}}\|$. Examples are discussed which confirm the above finding.

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